On the second-order asymptotics for entanglement-assisted communication

Abstract

The entanglement-assisted classical capacity of a quantum channel is known to provide the formal quantum generalization of Shannon’s classical channel capacity theorem, in the sense that it admits a single-letter characterization in terms of the quantum mutual information and does not increase in the presence of a noiseless quantum feedback channel from receiver to sender. In this work, we investigate second-order asymptotics of the entanglement-assisted classical communication task. That is, we consider how quickly the rates of entanglement-assisted codes converge to the entanglement-assisted classical capacity of a channel as a function of the number of channel uses and the error tolerance. We define a quantum generalization of the mutual information variance of a channel in the entanglement-assisted setting. For covariant channels, we show that this quantity is equal to the channel dispersion and thus completely characterize the convergence toward the entanglement-assisted classical capacity when the number of channel uses increases. Our results also apply to entanglement-assisted quantum communication, due to the equivalence between entanglement-assisted classical and quantum communication established by the teleportation and super-dense coding protocols.

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Acknowledgments

We are especially grateful to Milan Mosonyi for insightful discussions and for his help in establishing the proof of Propositions 11 and 14. We acknowledge discussions with Mario Berta, Ke Li, Will Matthews, and Andreas Winter, and we thank the Isaac Newton Institute (Cambridge) for its hospitality while part of this work was completed. MT is funded by the Ministry of Education (MOE) and National Research Foundation Singapore, as well as MOE Tier 3 Grant “Random numbers from quantum processes” (MOE2012-T3-1-009). MMW acknowledges startup funds from the Department of Physics and Astronomy at LSU, support from the NSF through Award No. CCF-1350397, and support from the DARPA Quiness Program through US Army Research Office award W31P4Q-12-1-0019.

Appendix 1: Heisenberg–Weyl operators

For any \(x,z \in \{0,1,\ldots , d\}\) the Heisenberg–Weyl Operators X(x) and Z(z) are defined through their actions on the vectors of the qudit computational basis \(\{|j\rangle \}_{j \in \{0,1,\ldots , d-1\}}\) as follows:

is the type of \(x^n\) and \(D(t\Vert q):= \sum _{a \in \mathcal {X}} t(a) \log \frac{t(a)}{q(a)}\) is the Kullback-Leibler divergence of the probability distributions t and q. From (7.1), (7.3) and (7.4) it follows that for any sequence \(x^n \in \mathcal {X}^n\) of type t,